Faraday, Maxwell, and the Electromagnetic Field

by Nancy Forbes

  Mathematics had a long and illustrious history at Cambridge, starting from when Isaac Newton held the Lucasian Chair of Mathematics. Cambridge was the place where one studied mathematics, while Oxford's genius lay more in the humanities. From the mid-1700s onward, Cambridge had developed its highly competitive examination culture, in which highly coached students strove to be ranked as high as possible in the Tripos listing. The system produced not only some of the best British mathematicians and scientists of the nineteenth century but also eminent churchmen, doctors, lawyers, civil servants, and economists. Success in the high-pressure Tripos was held to be a sign of a first-rate mind, able to work under trying conditions and solve problems in any field. Oxford, by contrast, largely eschewed competition, perhaps thinking it an unseemly activity for gentlemen; honors classifications there were never individually ranked—nor were all students expected to study mathematics.

  The Tripos required virtuosity in solving the set problems quickly and accurately. These problems usually had little connection with reality—they were contrived puzzles that demanded mastery of a wide repertoire of tricks and shortcuts. This was not Maxwell's forte, so he set out to master these skills. It was routine for all students aiming at high honors to be coached by one of the freelance private tutors whose income depended on results, and Maxwell joined the class of William Hopkins, the renowned “wrangler maker.” Hopkins was a driver but not a dull one, as one of his old students, the anthropologist Francis Galton, recalled:

  Hopkins to use a Cantab expression is a regular brick, tells funny stories connected with different problems, and is in no way Donnish, he rattles at a splendid pace and makes mathematics anything but a dry subject. I never enjoyed anything so much before.

  Hopkins had coached more than two hundred wranglers4 but had never met anyone like Maxwell, whom he described as “unquestionably the most extraordinary man I have ever met with in the whole of my experience.” “It is not possible,” he said, “for Maxwell to think incorrectly on physical subjects; in his analysis, however he is far more deficient.” It was Hopkins's job to subject Maxwell's freewheeling mind to what Forbes had called the “drill” of Cambridge. P. G. Tait, who had also studied under Hopkins, described the task that faced the tutor, and the manner in which his friend responded to drilling.

  [Maxwell] brought to Cambridge in the autumn of 1850 a mass of knowledge which was really immense for so young a man but in a state of order appalling to his methodical private tutor. Though the tutor was William Hopkins, the pupil to a great extent took his own way, and it may safely be said that no high wrangler of recent years ever entered the Senate House more imperfectly trained to produce “paying” work than did Clerk Maxwell.

  Maxwell did, however, learn much from Hopkins—he came to see the advantages of attacking problems systematically and the usefulness of the standard algebraic procedures. By applying routine drills and checks he was able to reduce his tendency to make algebraic mistakes, though this remained a weakness—“I am quite capable of writing a fancy formula,” he used to say, meaning a wrong one. Like Faraday, he liked to picture a problem, and on at least one occasion when Hopkins had filled the blackboard with equations, he solved the problem in a few lines with a diagram. Though he never “crammed” in the way that was thought necessary for high honors in the Tripos, he dutifully carried out all the set work. A fellow student, W. N. Lawson, reports:

  Maxwell was, I daresay you remember, very fond of a talk upon almost anything. He and I were pupils (at an enormous distance apart) of Hopkins, and I well recollect how, when I had been working all the night before and all the morning at Hopkins’ problems with little or no result, Maxwell would come in for a gossip and talk on and on while I was wishing him far away, till at last about half an hour before our meeting at Hopkins’, he would say—“Well, I must go to old Hop's problems”; and by the time we met they were all done.

  A few months before the Tripos, Maxwell became an observer in a widely debated university controversy. F. D. Maurice, a former student at Trinity and member of the Apostles, had gone on after leaving Cambridge to found the Christian Socialist movement, which had attracted a following among Maxwell's fellow students. The movement aimed to counteract the dehumanizing effects of industrial work by setting up cooperatives and working-men's colleges. Maurice's Theological Essays had caused a stir in 1853 by seeming to question some of the Articles of the Church of England, and as a result he was summarily fired from his professorship at King's College, London. Maxwell, along with some of his friends, was appalled at Maurice's treatment, especially since he firmly supported the idea of education for the working man. Maxwell had already helped young farmworkers at Glenlair by lending them books from the family library, and later, as a fellow at Cambridge; and as a professor at Aberdeen and at King's College, London, he gave up an evening a week to teach at working-men's colleges. He clearly felt, along with Faraday, that the common man was capable of appreciating science and had a right to know about it.

  He didn't let up in his non-Tripos activities and, perhaps, overdid things. On a vacation with a friend's family in Suffolk, he developed a fever and became delirious. The family nursed him for two weeks and sent daily reports to his father. Much moved, and profoundly grateful for their kindness, Maxwell nevertheless couldn't help making a critical observation on their way of life. Everyone was so anxious to know and to heed everyone else's wishes that no one had a life of his own. How much better it would be, Maxwell thought, for everyone to be allowed some interests that were not to be too much encouraged or inquired into by the others—in his view, this would greatly increase the resources of the family as a whole.

  Exam time came, and he sat with the others day after day in the Senate House, at his father's suggestion wrapping his feet in a bit of blanket for warmth. In the evenings they all needed to relax, and a merry crowd gathered in Maxwell's rooms to dabble in experiments with magnets under his guidance. The results soon followed. E. J. Routh was announced as senior wrangler and J. Clerk Maxwell was placed second. The best students then competed for the Smith's Prize, and here Routh and Maxwell were declared joint winners. Routh was an exceptional mathematician who went on to do outstanding research. He did much to systematize the mathematical theory of mechanics; he created several ideas that found application in modern control theory; and, like the giants Laplace, Lagrange, and Hamilton, he gained the rare distinction of having a function named after him, the Routhian.5 He also took up coaching others for the Tripos and became the supreme “wrangler maker,” surpassing even Hopkins. Maxwell had done well, not quite as well as his friend Tait, who had been senior wrangler two years earlier, but Tait had not been up against Routh. He had established his mathematical credentials and put himself in a good position to gain a fellowship at Trinity. His father was delighted at the results, and congratulations tumbled in from uncles, aunts, and cousins.

  Maxwell's undergraduate years were joyful ones. And productive—he had completed the Tripos grind with honor and, thanks to Hopkins's drilling, his mathematics now had the discipline and poise that Forbes had earlier brought to Maxwell's experimental work. Society had played its part, too, by softening his eccentricities: he now impressed strangers as an interesting young man rather than merely an odd one. Best of all, he had made a set of lifelong friends. They included H. M. Butler, who went on to be headmaster of Harrow School and then master of Trinity College; F. W. Farrar, who became dean of Canterbury but is best remembered as the author of the popular moral tale Eric, or Little by Little; and R. B. Litchfield, who founded the London Working Men's College. The value that Maxwell put on friendship is evident from a letter he wrote later to Litchfield when another friend, Robert Henry Pomeroy, died in the Indian Rebellion of 1857.

  It is in personal union with my friends that I hope to escape the despair which belongs to the contemplation of the outward aspect of things with human eyes. Either be a machine and see nothing but “phenomena” or else try to be a man, feeling yo
urself interwoven, as it is, with many others and strengthened by them whether in life or death.

  Lewis Campbell gives us a picture of Maxwell as his friends saw him:

  His presence had by this time fully acquired the unspeakable charm for all who knew him which made him insensibly become the center of any circle, large or small, consisting of his friends or kindred.

  The society of Cambridge had done him good, as Forbes had predicted, but Maxwell had more than repaid the debt. A fellow student told Campbell:

  Of Maxwell's geniality and kindness of heart you will have had many instances. Everyone who knew him at Trinity can recall some kindness or some act of his which left an ineffaceable impression on the memory—for “good” Maxwell was in the best sense of the word.6

  After consulting his father, Maxwell decided to stay at Trinity as a bachelor scholar and apply for a fellowship. This wasn't a long-term plan—in those days, fellows of Trinity were required to be ordained into the Church of England within seven years and to remain unmarried, and he had no intention of making either commitment—but it did mean that he could now take up the ideas for scientific investigations that had been brewing in the back of his mind. He wanted to come to grips with electricity, but at this time the most pressing question on his mind was, how do we see colors? The way he answered it gives us an insight into his boldness, ingenuity, and resolution.

  When Maxwell was three, someone had said, “look at that lovely blue stone,” and he had responded, “but how d'ye know it's blue?”1 The question had still not been answered. In the early 1800s, Thomas Young had put forward the interesting idea that the human eye has three types of receptors, each sensitive to a particular color, and that the brain combines the signals to form a single perceived color. But Young couldn't supply any supporting evidence and his theory had been largely neglected for the best part of half a century when James Forbes thought of taking a disc with differently colored sectors, like a pie chart, and spinning it fast so that one sees not the individual colors but a blurred-out mix. His idea was that each of the eye's three types of receptors might respond to one of the primary colors used by artists—red, yellow, and blue—so he tried various mixtures of these colors on the spinning disc to see what combined color would appear. The results were puzzling. When, for example, he mixed just yellow and blue, he didn't get green, as painters did, but a dull sort of pink. And he couldn't get white, no matter how he mixed the colors.

  This was as far as Forbes got, but Maxwell took up the idea and soon discovered the source of his mentor's confusion. Forbes had failed to distinguish between mixing colors in the light that reaches the eye, as when spinning a disc, and mixing pigments, as a painter does. Pigments extract color from light—what you see is whatever light is left over after the pigments have done their extraction. So perhaps Forbes's choice of the painter's primary colors, red, yellow, and blue, was wrong. Maxwell tried, instead, mixing red, green, and blue, and the outcome was spectacular. Not only did he get white by using equal proportions of the three colors, but he found that he could produce a great variety of colors, also, simply by varying the proportions of red, green, and blue.

  To put things on a proper footing, he ordered sheets of colored paper in many colors from an Edinburgh printer and had a special disc made—he called it his “color top.” About six inches in diameter, it had percentage markings around the rim, a handle, and a shank for winding a pull-string. He cut out paper discs of red, green, and blue and slit them so that they could be overlapped on the color top with any desired amount of each color showing. This way, he was able to measure what percentages of red, green, and blue on the spinning disc matched the color of whatever paper was held alongside the spinning color top for comparison. But then he thought of a better arrangement. Instead of holding a separate piece of paper alongside, he put a smaller paper disc of the color he wanted to match on top of the red, green, and blue ones, so that it occupied the inner part of the top's disc. With this arrangement, he could add a sector of black, if needed, to be able to match brightness as well as hue.

  Using this homely device, he showed that you really can get any color you want by mixing red, green, and blue in the right proportions—exactly the principle used in our television sets today. Maxwell got all of his friends and colleagues to have a go at mixing colors, and he found remarkably little variation in color perception among people with normal vision. He particularly sought out color-blind people and found that most of them lacked fully functioning red-sensitive receptors, which explained their difficulty in telling red from green.

  All of this was groundbreaking work, but there was no fanfare: Maxwell simply sent off a paper to the Royal Society of Edinburgh and demonstrated some of the results to the Cambridge Philosophical Society using his color top. In his own view, the work done so far was no more than a preliminary sketch because the colors of the printer's paper were arbitrary—simply “specimens of different kinds of paint.”2 To get precise and replicable results, he needed to use pure spectral colors extracted from sunlight, so he devised a “color box” to do this, using a prism to spread the sunlight into a spectrum, adjustable slits to select particular colors, and further optical arrangements to combine the selected colors. Over the years, he made several versions of the color box, improving it each time, and the work became a lifelong project. Had he done nothing else, he would now be known as Maxwell, one of the great founders of the science of color vision.3

  While spinning the top, he had been fulfilling his duties as a bachelor scholar, giving classes and supervising examinations. The role wasn't onerous, but he took it seriously and volunteered to take extra classes, which was good practice for a professorship later on. He strongly supported the movement led by F. D. Maurice for working-men's colleges and began the practice that he continued in Aberdeen and London of giving up an evening a week to talk at the local college. Friendships multiplied, and his social life was full. He continued his association with the Apostles and was elected to the exclusive Ray Club, a forum for discussing and promoting the natural sciences. As for exercise, there was walking, rowing on the Cam, jumping and vaulting in the gymnasium, and swimming in the new pool, where he helped to organize group sessions to make things more sociable. As if this were not enough, he kept up his formidable regime of general reading, taking in, among others, Carlyle, Chaucer, Francis Bacon, Pope, Goldsmith, Berkeley, and Cowper.

  On a parallel track, his thoughts turned more and more to electricity and magnetism. Years of lighthearted experimenting—copper-plating jam jars, playing with magnets, building model telegraphs—had given him a fascination for the subject, and the time had come to begin a serious study. However, it wasn't at all clear where to start; he needed advice, and, through a family connection, he knew exactly where to find it.

  His cousin Jemima had married Hugh Blackburn, a professor of mathematics at Glasgow University. The professor of natural philosophy there, and Blackburn's closest friend, was none other than William Thomson, who, as we have seen, was one of the few scientists to have taken Faraday's idea of lines of force seriously. Maxwell had met Thomson several years earlier when visiting Jemima with his father. Like everyone, he was impressed by the dashing young man of science, and the feeling was mutual. Maxwell wrote to Thomson from Cambridge, breezily announcing his intention “to poach on your electrical preserves” and asking for a reading list. Thomson was delighted to take on the role of mentor—he had many other interests and was now becoming involved in the great Atlantic telegraph-cable project.4 His reply to Maxwell hasn't survived, but we can be sure that Faraday's Experimental Researches in Electricity had a prominent place on the reading list.

  Scanning the books and papers that Thomson had recommended, Maxwell soon saw that the state of knowledge about electricity and magnetism was unsatisfactory. Much had been written, but each leading author had his own methods, terminology, and point of view. All the theories except Faraday's were mathematical and based on the idea of action at a
distance. Their authors had largely spurned Faraday's notion of lines of force because it couldn't be expressed in mathematical terms, except, in a limited way, through an analogy Thomson had made between electric lines of force and the steady flow of heat through a metal bar.

  Nevertheless, Maxwell was drawn to Faraday, both by Thomson's encouragement and by his own intuition—truth lay in observed results, and anyone aspiring to solve the remaining mysteries of electricity and magnetism should first study what had been found by experiment. He resolved to read all of Faraday's Researches before tackling the mathematical treatments. He was struck at once by Faraday's openness and integrity, and, as he read more, he came to see the intellectual strength of the work. Maxwell's experience from all the hours spent working in his improvised laboratory at Glenlair made him marvel not only at the precision of the great man's experiments but also at the power and subtlety of the reasoning that followed. To Maxwell, Faraday's ideas rang true: he had found a kindred spirit and a new source of inspiration. The warmth he already felt for Faraday is evident from a comment he later included in his Treatise on Electricity and Magnetism:

  The method which Faraday employed in his researches consisted in a constant appeal to experiment as a means of testing the truth of his ideas, and a constant cultivation of ideas under the direct influence of experiment…. Faraday…shows us his unsuccessful as well as his successful experiments and his crude ideas as well as his developed ones, and the reader, however inferior to him in inductive power, feels sympathy even more than admiration, and is tempted to believe that, if he had the opportunity, he too would be a discoverer.5